[sdiy] analog through zero FM dummy

[Martin Czech]

Ok, back to sanity after a nice wisdom tooth extraction...
(I'm very suspicious, is this translation ok?)
I have another one, so another week of "fun" is waiting for me...

Michael B.'s list archive is great. I can now dig into all those
threads...

There was a thread started by Rene about TZFM. It all started with the
fact that switching triangle polarity due to modulator and switching due
to level can interact in a messy way and then came several attempts to
solve the conflict. So far, so good, but now I don't understand the whole
concept of this kind of modulation any more, I mean: is this really FM?

Let me see if I can get it together:

we start with a tri wave:

tri(p(t)) 

where p(t) is our phase function. Phase relates to charge
in this kind of oscillators.

We want to write down everthing in terms of frequency, because that
would relate to current, makes things easier.
We exploit 

p(t)=INT{f(t)}dt +p0

simply stating that phase is integrated frequency information,
or OTOH charge is integrated current. See if it works for 
f(t)=wc=const:

tri(wc*t)

ok, that seems to be wellknown.

Now some modulation:

p(t)=INT{wc+M*sin(wm*t)}dt +p0

This would be the right formula for sinuoidal frequency modulation
with some modulation frequency wm arround some carrier wc.  I hope you
all aggree. We can put that plain into our tri wave, forgetting about
some signs:

tri(wc*t+M/wm*cos(wm*t)+p0)

I hope you're still with me. The point is: the modulation term simply
ADDS to the running carrier phase.

OTOH, what do the known FM circuits do? They should ADD some current
into the timing cap, this would mean another current source side by
side to the normal expo current source. But all the circuits I know
seem to influence the REFERENCE current of the expo source instead.
This gives a current like:

i(t)=(i0+M*sin(wm*t))*exp{C*U/Ut}

or:

i(t)=i0*exp{C*U/Ut}+M*sin(wm*t)*exp{C*U/Ut}

this is corresponding to our frequency function.  My point is now that
the modulation index M is also scaled with exp{C*U/Ut}, ie. if the 
carrier frequnency gets higher, so does effective M. This seems not to be the
same as in the theoretical case above which should have translated
into:

i(t)=i0*exp{C*U/Ut}+M*sin(wm*t)


Or speaking electronic wise: 

not the reference current should be modulated, instead a linear
current should be additionally injected into the timing cap.
This would mean another ota in most cases. Not that it would help much
with our problems we have with the usual tri oscillator due to polarity
switching...
btw. Rene's tricky circuit follows the same path in this concern.
The multiplication happens in the ota...

So I claim: analog TZFM is not the same as "theoretical FM".

???

m.c.